TITLE: Why are primes considered to be the "building blocks" of the integers?
QUESTION [63 upvotes]: I watched the video of Terence Tao on Stephen Colbert the other day (here), and he stated, like many mathematicians do, that the primes are the building blocks of the integers.
I've always had trouble with this idea, because I don't think it's a philosophically sound idea.
I'm going to start by stating the basic argument: An integer can either be expressed as a product of smaller integers or not.  If not, it's called "prime".
But you could say the same thing about powers: An integer can be expressed as a (well, I'm not a mathematician and now I'm at a loss for words) power-chain of integers or not.  If not, it's called (I don't know what the name could be) "power-prime".  For example, 16 = 2^2^2, so it's power-composite, whereas 2 can't be expressed with powers, so is power-prime.
Again, you could say the same thing about addition: An integer can be expressed as a sum of integers or not.  If it can, it's called an integer.  If not, it's called "unit", I guess, because every integer can be expressed as a sum of 1's!  So, truly, the unit is the building block of the integers.
So, I don't think there's any truth in the idea that the primes are the building blocks of the integers. I think it's just an expression of a human preference for multiplication because multiplication is complicated enough to be interesting to us (unlike addition), but not so complicated that it's bewildering (like powers).  Or maybe because the notation we use to express powers doesn't use an explicit operator, nobody has noticed that you can treat powers analogously to multiplication in a situation like this.  Just a couple of theories.
I'd be curious to know what research has been done in this area.  I bet the power-primes have a lot of interesting properties.
Further clarification of what I mean by "power-prime":
I'm just talking about factoring integers using the exponentiation operator rather than the multiplication operator.
I apologize for using the ugly term "power-prime", but I don't know what these are officially called.  I don't even know what to call a bunch of numbers that are combined by exponentiation!  A bunch of numbers combined by addition is called a "sum".  Numbers combined by multiplication is called a "product".  What is the term for a number of numbers combined by exponentiation?  I'm having a major memory lapse because I feel I should know what it is.
I'll make a table because examples are easiest to understand.  I'm going to use the caret ^ character for exponentiation (2^2 = $2^2$).
n   factoring  name
1
2              p-prime
3              p-prime
4   2^2        p-composite
5              p-prime
6              p-prime
7              p-prime
8   2^3        p-composite
9   3^2        p-composite
10             p-prime
11             p-prime
12             p-prime
13             p-prime
14             p-prime
15             p-prime
16  2^2^2      p-composite
17             p-prime
18             p-prime
19             p-prime
20             p-prime
21             p-prime
22             p-prime
23             p-prime
24             p-prime
25  5^2        p-composite
26             p-prime
27  3^3        p-composite
28             p-prime
29             p-prime
30             p-prime
31             p-prime
32  2^5        p-composite

See, every natural number is either p-prime or a combination of p-primes (via exponentiation).  I don't know if each of these factorings is unique, since, as was mentioned by Pete, it's highly non-obvious.  I honestly don't think that Fundamental Theorem of Arithmetic helps here because exponentiation is not commutative, but I could be wrong.  It's also noteworthy that our usual exponentiation operator is right-associative.  A left-associative exponent would give a slightly different table, since for example (2^2)^3 $\ne$ 2^(2^3).

REPLY [3 votes]: Kaplan and Kaplan's The Art of the Infinite (fantastic read, by the way) touches on this. Indeed, it goes into some depth about the sheer leap from 1 as the sole additive "building block", to all the primes being the multiplicative "building blocks". (I don't recall if the term "building block" was used, but the notion is there.)
I can't speak for Terry Tao or other mathematicians, but I would mean there to be an implicit "multiplicative" (as in the previous paragraph) if and when I called the primes the building blocks of the integers. In that sense it is wholly correct.
Your question seems to acknowledge this and go on to ask why it's multiplication that's implied. You're probably right about multiplication being "complicated enough to be interesting to us", though not, I think, about powers being "bewildering". "Building" integers through addition is trivial, and so multiplication is merely the first non-trivial operation we come to. (First, in the sense of multiplication being "meta-addition", powers being "meta-multiplication", etc.)